Nowadays, the nano-machining process is used to produce high quality finished surfaces with precise form accuracy. To understand and analyze the chip formation mechanism of nano-machining process on an atomistic scale, since the experimentation is not an easy task, numerical simulation such as molecular dynamic (MD) simulation is a very useful method. In this paper, MD simulation of the nano-metric cutting of single-crystal copper was performed with a single crystal diamond tool. The model was solved with both pair wise Morse potential function and embedded atom method (EAM) potential to simulate the inter-atomic force between the work-piece and a rigid tool. The chip formation mechanism, dislocation generation, tool forces and generated temperature were investigated. Results show that the Morse potential cannot perform an appropriate defect formation and plastic deformation in nano-metric cutting of metals. Also, tool forces in Morse potential are more than the forces in EAM potential. Furthermore, the fluctuations of resultant forces in Morse potential are greater than that of EAM. In addition, using many-body interaction potentials like EAM can lead to substantial changes in surface energies, elastic-plastic properties and atomic displacement, compared with the pair-wise potentials like Morse. Finally, the atomic displacement investigation shows that in EAM potential study, only the atoms in a local region near the cutting process are displaced, but in Morse potential a large portion of atoms has affected during cutting process. Subsequently, the chip temperature in EAM potential is more than that of Morse potential.